Right, I promised a response and here is my first thought. If Henry has a twin brother, he may well not be brought up quite as spoiled in the first place, which would impact on his personality anyway. Think about it. In his childhood, Henry was brought up very much as the little lord of Eltham Palace. He was the only son in the household, at least barring a few months when Edmund, Duke of Somerset was around, and IIRC, seems to have been Margaret Beaufort's favourite grandchild, as far as she had such a thing. It wouldn't surprise me if EOY wasn't fond of him because he reminded her of her father either. He was probably brought up in quite an indulgent fashion, even if he was made to realise that he would never be more than Arthur's aide-de-camp and Duke of York rather than King. Having a twin brother would probably change that quite a bit. He wouldn't be the only boy in the nursery, for one.
To quote Frances Hodgson Burnett's 'The Secret Garden'
Henry would be in a similar situation, I suspect. Having a brother to tussle with and fight for attention with on a daily basis might actually mean he grew up more even-keeled in the long run.